Hyperfiniteness of boundary actions of acylindrically hyperbolic groups

Abstract

We prove that for any countable acylidrically hyperbolic group $G$, there exists a generating set $S$ of $G$ such that the corresponding Cayley graph $\Gamma(G,S)$ is hyperbolic, $|\partial\Gamma(G,X)|>2$, the natural action of $G$ on $\Gamma(G,S)$ is acylindrical, and the natural action of $G$ on the Gromov boundary $\partial\Gamma(G,S)$ is hyperfinite. This result broadens a class of groups that admit a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.